{
  "abstract": "We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations $x \\mapsto x+α$ on $\\mathbb{T}$ in terms of the irrationality exponent $μ(α)$. For the binary coding determined by an interval $[0,β)$, let $d_N(x,y)$ be the edit distance between the codings of the orbits of the points $x$ and $y$ up to time $N-1$. For every $α\\notin \\mathbb{Q}$, almost every $β\\in (0,1)$, and almost every $x,y \\in \\mathbb{T}$, we determine the edit-distance exponents $\\displaystyle (*) \\; \\; \\limsup_{N\\to\\infty}\\frac{\\log d_N(x,y)}{\\log N} = \\frac{ρ(α)}{ρ(α)+1} \\; \\;$ and $\\; \\; \\displaystyle\\liminf_{N\\to\\infty}\\frac{\\log d_N(x,y)}{\\log N} = \\min\\Bigl\\{\\frac{1}{2},\\frac{1}{ρ(α)}\\Bigr\\}$, where $ ρ(α):=μ(α) - 1$. We find a surprising transition at the golden mean $φ= \\frac{1+\\sqrt{5}}{2}$: If $ ρ(α) \\le φ$, then both asymptotics in $(*)$ hold for all $β$. However, for every finite $ρ>φ$, there is an uncountable set of $α$ with $ρ(α)=ρ$ for which both edit-distance exponents are strictly smaller than the corresponding values in $(*)$ for uncountably many $β$. We also derive consequences for aperiodic circle homeomorphisms and Sturmian sequences. For rotations of $\\mathbb{T}^d$ coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is $d/(d+1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.11763",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrew Best"
    },
    {
      "name": "Yuval Peres"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vcqgvx5y02dm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.11763",
  "fulltext_url": null,
  "id": "7vcqgvx5y02dm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.11763",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7202342562",
  "pdf_url": "https://arxiv.org/pdf/2608.11763",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vcqgvx5y02dm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2608.11763",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.11763"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Edit distance exponents for irrational rotations",
  "title_text": "Edit distance exponents for irrational rotations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/7vcqgvx5y02dm",
  "venue": "arXiv",
  "versions": []
}